Atlas › Modular forms › Weight 4 › Level 42 › 42.4.a

Newspace 42.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(42))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
42.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} + 18q^{5} - 6q^{6} + 7q^{7} + O(q^{8})$	− + −	$+1$
42.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} + 2q^{5} + 6q^{6} - 7q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
42.4.a.a	$x + 198$
42.4.a.b	$x + 22$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(42))$	2
Old forms	18
Cusp forms $S_{4}(\Gamma_0(42))$	20
Eisenstein series $E_{4}(\Gamma_0(42))$	8
Modular forms $M_{4}(\Gamma_0(42))$	28

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(42)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.114 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 3, 6, 7, 14, 21, 42]: [1, 2, 2, 5, 3, 8, 10, 28] (certified at ell = 2147483647)
T = [(5, 1), (11, 3)]; h of degree 2 = 1 x 2 (phi(ord eps) x dim S_k^new); 1 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(42, 4)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 42:4 1000

← 41.4.a · 43.4.a →